Subventions et des contributions :

Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023)

This research program is concerned with the study of invariant subspaces of linear maps and their applications. If H is a Hilbert space and Y is a subspace of H, then Y is called invariant under a linear map T on H if T maps elements of Y back to Y. This is a classical idea, and there is a very large body of knowledge about invariant subspaces accumulated through last several decades.

In the first part of the program, we propose to study various similarity problems. Similarity problems are very natural problems in operator theory, and some of the most famous and difficult problems in operator theory are similarity problems. That methods of the invariant subspace research form a promising approach to these problems was demonstrated in a number of recent papers. In particular, amenability of many nonselfadjoint operator algebras is conveniently described in terms of invariant subspaces. The typical conclusion is that if the invariant subspaces of the algebra behave in a regular way (described in precise terms), then the algebra is similar to a C*-algebra, which is a much better understood object than an arbitrary operator algebra.

In this proposal, we produce a number of new conjectures about connections between invariant subspaces and amenability of operator algebras. We also propose to study important Banach algebras using methods of the invariant subspace research. We make conjectures about the following classical problem in the analysis of groups: when is every bounded representation of a discrete group G on B(H) similar to a unitary representation? We suspect that this can be described using the so-called total reduction property (TRP) defined in terms of invariant subspaces. We hope to obtain quantitative and combinatorial conditions on the group that ensure the TRP. In addition, we propose to study classical Banach algebras, such as the algebra B(X), having the TRP.

In the second part of this program we will study specific questions about existence and special properties of invariant subspaces for operators. In particular, we hope to find properties of invariant subspaces of the shift operator on sequence spaces. Next, we intend to study invariant subspaces of perturbations of operators, a topic that has been on the sharp rise in the last few years. We will also look at positive operators on Banach lattices and specific transitive algebras. Other conjectures will also be considered.

Finally, we will study certain preserver problems in Matrix analysis using ideas of the invariant subspace research.